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Beliebt Rechnen >

integral of (ae^θ+b)/(ae^θ-b)

Lösung

\int \frac{a e ^{θ} + b}{a e ^{θ} - b} d θ

Lösung

- ln (\frac{e ^{θ} ∣ a ∣}{∣ b ∣}) + 2 ln \frac{a e ^{θ}}{b} - 1 + C

Schritte zur Lösung

\int \frac{a e ^{θ} + b}{a e ^{θ} - b} d θ

b + a e^{θ} = b (1 + \frac{a e ^{θ}}{b})

= \int \frac{b ( 1 + \frac{a e ^{θ}}{b} )}{- b + a e ^{θ}} d θ

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= b \cdot \int \frac{1 + \frac{a e ^{θ}}{b}}{- b + a e ^{θ}} d θ

Wende U-Substitution an

= b \cdot \int \frac{u}{b ( u - 1 ) ( u - 2 )} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= b \frac{1}{b} \cdot \int \frac{u}{( u - 1 ) ( u - 2 )} d u

Ermittle den Partialbruch von

\frac{u}{( u - 1 ) ( u - 2 )} : - \frac{1}{u - 1} + \frac{2}{u - 2}

= b \frac{1}{b} \cdot \int - \frac{1}{u - 1} + \frac{2}{u - 2} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= b \frac{1}{b} (- \int \frac{1}{u - 1} d u + \int \frac{2}{u - 2} d u)

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

\int \frac{2}{u - 2} d u = 2 ln ∣ u - 2 ∣

= b \frac{1}{b} (- ln ∣ u - 1 ∣ + 2 ln ∣ u - 2 ∣)

Setze in

u = 1 + \frac{a e ^{θ}}{b}

ein

= b \frac{1}{b} (- ln 1 + \frac{a e ^{θ}}{b} - 1 + 2 ln 1 + \frac{a e ^{θ}}{b} - 2)

Vereinfache

b \frac{1}{b} (- ln 1 + \frac{a e ^{θ}}{b} - 1 + 2 ln 1 + \frac{a e ^{θ}}{b} - 2) : - ln (\frac{e ^{θ} ∣ a ∣}{∣ b ∣}) + 2 ln \frac{a e ^{θ}}{b} - 1

= - ln (\frac{e ^{θ} ∣ a ∣}{∣ b ∣}) + 2 ln \frac{a e ^{θ}}{b} - 1

Füge eine Konstante zur Lösung hinzu

= - ln (\frac{e ^{θ} ∣ a ∣}{∣ b ∣}) + 2 ln \frac{a e ^{θ}}{b} - 1 + C

Graph

Beliebte Beispiele

y= 1/(1+e^{-x)}

y = \frac{1}{1 + e ^{- x}}

(\partial)/(\partial x)(e^{2x}sin(2y))

\frac{\partial}{\partial x} (e^{2 x} sin (2 y))

limit as x approaches 2 of (x^2-4)/(x+2)

x \to 2 lim (\frac{x ^{2} - 4}{x + 2})

2x^{''}-3x^'+x=0

2 x^{^{''}} - 3 x^{^{'}} + x = 0

derivative of ((1/x+1/(x^2))/(x-1))

\frac{d}{d x} (\frac{( \frac{1}{x} + \frac{1}{x ^{2}} )}{x - 1})